Necessary Conditions for the Realizability of n-Port Resistive Networks with more than (n + 1) Nodes

Abstract

A new set of necessary conditions are presented for an nth-order square symmetric matrix Y with real entries to be the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port admittance matrix of a network containing positive resistors and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 2)</tex> nodes. The so called "supremacy" conditions represent a symmetric set of inequalities among products of pairs of elements in a matrix S obtained in a simple manner as a linear combination of entries in Y. The conditions are derived by augmenting the given matrix Y with an additional port to form a connected port structure, transforming the port structure until it is linear, applying the "uniform tapering conditions" and then eliminating the augmenting port. Extensions are given to the case of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n + p</tex> nodes with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2 &lt;P \leq n.</tex>